Question

Given two odd intgers a and b prove that a^3− b^3 is divisible by 2^n if and only if a-b is divisible by 2^n.​

Answer 1

Step-by-step explanation:

We have

a and b are two odd positive integers such that a & b

but we know that odd numbers are in the form of 2n+1 and 2n+3 where n is integer.

so, a=2n+3, b=2n+1, n∈1

Given ⇒ a>b

now, According to given question

Case I:

2

a+b

=

2

2n+3+2n+1

=

2

4n+4

=2n+2=2(n+1)

put let m=2n+1 then,

2

a+b

=2m ⇒ even number.

Case II:

2

a−b

=

2

2n+3−2n−1

2

2

=1 ⇒ odd number.

Hence we can see that, one is odd and other is even.

This is required solutions.

Answer 2

Answer:

We have

a and b are two odd positive integers such that a & b

but we know that odd numbers are in the form of 2n+1 and 2n+3 where n is integer.

so, a=2n+3, b=2n+1, n∈1

Given ⇒ a>b

now, According to given question

Case I:

2

a+b

=

2

2n+3+2n+1

=

2

4n+4

=2n+2=2(n+1)

put let m=2n+1 then,

2

a+b

=2m ⇒ even number.

Case II:

2

a−b

=

2

2n+3−2n−1

2

2

=1 ⇒ odd number.

Hence we can see that, one is odd and other is even.

This is required solutions.

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